Birman anticipates the development of an algorithm which will begin with two oriented links L and L', represented by closed n and r-braids respectively (or alternatively by arbitrary link diagrams with n and r Seifert circles respectively), and which will decide whether these links are isotopic. For the special case where L' is the standard representative of the r- component unlink, she expects that the algorithm will be a polynomial-time algorithm. Morgan plans to study compactness questions for families of hyperbolic n-manifolds, and more generally the problem of compactifying such families. He intends to study the C-infinity classification of algebraic surfaces using new ideas introduced by Donaldson involving gauge theory. Hodgson's research deals mostly with the investigation of 3- dimensional manifolds using geometric methods. His main projects will involve the study of the hyperbolic Dehn surgery spaces introduced by Thurston. By using techniques including Hodge theory and analysis of geometric limits he hopes to obtain new results on the local and global structure of these Dehn surgery spaces. This would lead to new methods for constructing geometric structures on certain 3-manifolds, for example surface bundles over the circle. He also intends to continue his investigations into the variation of geometry as hyperbolic structures are deformed, especially the kinds of degeneration that can occur: including degeneration to lower dimensional limits and splitting along incompressible surfaces. This will include some computational studies involving the construction of geometric structures on 3-manifolds. He will also study the new invariants of 3-manifolds defined recently by Casson and Floer.
